We present the first study of the far-infrared (FIR) properties of high-redshift, radio-selected ultraluminous infrared galaxies (ULIRGs) using deep observations obtained with the Spectral and Photometric Imaging Receiver (SPIRE) from the Herschel Multi-tiered Extragalactic Survey (HerMES). These galaxies span a large range of 850-μm fluxes from submillimetre-luminous ~10mJy sources (SCUBA galaxies) to ~1.5mJy from stacked SCUBA non-detections, thus likely representing a complete distribution of ULIRG spectral energy distributions (SEDs). From Keck spectroscopic surveys in the Lockman-North field we identified a sample of 31 submillimetre galaxies (SMGs) and 37 submillimetre-faint, optically faint radio galaxies (OFRGs), all with radio-inferred IR luminosities >1012 Lsolar. These galaxies were cross-identified with SPIRE 250-, 350- and 500-μm catalogues based on fluxes extracted at 24-μm positions in the SWIRE survey, yielding a sample of more than half of the galaxies well detected in at least two of the SPIRE bandpasses. By fitting greybody dust models to the SPIRE photometry together with SCUBA 850-μm measurements (for OFRGs, only 850- upper limits), we infer dust temperatures and FIR luminosities. The OFRGs detected by SPIRE have median = 41 +/- 5K and the SMGs have = 34 +/- 5K, both in reasonable agreement with previous (pre-Herschel) estimates, reaffirming that the local FIR/radio correlation holds (at least for this subset of high-z ULIRGs) at high redshift (we measure = 2.43 +/- 0.21 using SIR derived from greybody fit coupled with a power-law extrapolation to the 24 μm). Our observations first confirm that a substantial fraction of OFRGs exhibits large infrared luminosities corresponding to SFRs of ~400Msolaryr-1. The SPIRE observations secondly confirm the higher dust temperatures for these OFRGs than similarly selected SMGs, consistent with early predictions of the submillimetre-faint radio populations. Our observations also clearly confirm the large infrared luminosities of most SMGs selected with S850μm > 5mJy and radio and strong 24-μm detections, corresponding to SFRs of ~700Msolaryr-1.